(ii) Given that gðxþ ¼ð4x 1Þðax 2 þ bx þ cþ, find the values of the integers a, b and c. (3 marks)

Centre Number Candidate Number For Examiner s Use Surname Other Names Candidate Signature Examiner s Initials Mathematics Unit Pure Core 4 Tuesday 15 June 2010 General Certificate of Education Advanced Level Examination June 2010 9.00 am to 10.30 am For this paper you must have: the blue AQA booklet of formulae and statistical tables. You may use a graphics calculator. Time allowed 1 hour 30 minutes MPC4 Instructions Use black ink or black ball-point pen. Pencil should only be used for drawing. Fill in the es at the top of this page. Answer all questions. Write the question part reference (eg (a), (b)(i) etc) in the left-hand margin. You must answer the questions in the spaces provided. around each page. Show all necessary working; otherwise marks for method may be lost. Do all rough work in this book. Cross through any work that you do not want to be marked. Information The marks for questions are shown in brackets. The maximum mark for this paper is 75. Advice Unless stated otherwise, you may quote formulae, without proof, from the booklet. Question 1 2 3 4 5 6 7 8 TOTAL Mark (JUN10MPC401) 6/6/6/ MPC4

2 Answer all questions in the spaces provided. 1 (a) The polynomial fðxþ is defined by fðxþ ¼8x 3 þ 6x 2 14x 1. Find the remainder when fðxþ is divided by ð4x 1Þ. (2 marks) (b) The polynomial gðxþ is defined by gðxþ ¼8x 3 þ 6x 2 14x þ d. (i) Given that ð4x 1Þ is a factor of gðxþ, find the value of the constant d. (2 marks) (ii) Given that gðxþ ¼ð4x 1Þðax 2 þ bx þ cþ, find the values of the integers a, b and c. (3 marks) (02)

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4 2 A curve is defined by the parametric equations x ¼ 1 3t, y ¼ 1 þ 2t 3 (a) Find dy dx in terms of t. (3 marks) (b) Find an equation of the normal to the curve at the point where t ¼ 1. (4 marks) (c) Find a cartesian equation of the curve. (2 marks) (04)

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6 3 (a) (i) Express (ii) Hence find 7x 3 ðx þ 1Þð3x 2Þ (b) Express 6x2 þ x þ 2 2x 2 x þ 1 ð in the form A 7x 3 ðx þ 1Þð3x 2Þ dx. in the form P þ Qx x þ 1 þ B 3x 2. þ R 2x 2 x þ 1. (3 marks) (2 marks) (3 marks) (06)

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8 3 4 (a) (i) Find the binomial expansion of ð1 þ xþ 2 up to and including the term in x 2. (2 marks) (ii) 3 Find the binomial expansion of ð16 þ 9xÞ 2 up to and including the term in x 2. (3 marks) 3 (b) Use your answer to part (a)(ii) to show that 132 46 þ a, stating the values of the b integers a and b. (2 marks) (08)

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10 5 (a) (i) Show that the equation 3 cos 2x þ 2 sin x þ 1 ¼ 0 can be written in the form 3 sin 2 x sin x 2 ¼ 0 (3 marks) (ii) Hence, given that 3 cos 2x þ 2 sin x þ 1 ¼ 0, find the possible values of sin x. (2 marks) (b) (i) (ii) Express 3 cos 2x þ 2 sin 2x in the form R cosð2x aþ, where R > 0 and 0 < a < 90, giving a to the nearest 0.1. (3 marks) Hence solve the equation 3cos2x þ 2 sin 2x þ 1 ¼ 0 for all solutions in the interval 0 < x < 180, giving x to the nearest 0.1. (3 marks) (10)

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12 6 A curve has equation x 3 y þ cosðpyþ ¼7. (a) Find the exact value of the x-coordinate at the point on the curve where y ¼ 1. (2 marks) (b) Find the gradient of the curve at the point where y ¼ 1. (5 marks) (12)

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14 7 The point A has coordinates ð4, 3, 2Þ. The line l 1 passes through A and has equation r ¼ 4 2 3 2 1 The line l 2 has equation r ¼ 4 3 5 þ m4 4 The point B lies on l 2 where m ¼ 2. 3 1 5. 2 1 2 3 2 3 4 2 5 þ l4 0 5. 1 3 2 (a) ƒ! Find the vector AB. (3 marks) (b) (i) Show that the lines l 1 and l 2 intersect. (4 marks) (ii) The lines l 1 and l 2 intersect at the point P. Find the coordinates of P. (1 mark) (c) The point C lies on a line which is parallel to l 1 and which passes through the point B. The points A, B, C and P are the vertices of a parallelogram. Find the coordinates of the two possible positions of the point C. (4 marks) (14)

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16 8 (a) Solve the differential equation 1 dx dt ¼ 1 5 ðx þ 1Þ 2 given that x ¼ 80 when t ¼ 0. Give your answer in the form x ¼ fðtþ. (6 marks) (b) A fungus is spreading on the surface of a wall. The proportion of the wall that is unaffected after time t hours is x%. The rate of change of x is modelled by the differential equation 1 dx dt ¼ 1 5 ðx þ 1Þ 2 At t ¼ 0, the proportion of the wall that is unaffected is 80%. Find the proportion of the wall that will still be unaffected after 60 hours. (2 marks) (c) A biologist proposes an alternative model for the rate at which the fungus is spreading on the wall. The total surface area of the wall is 9 m 2. The surface area that is affected at time t hours is A m 2. The biologist proposes that the rate of change of A is proportional to the product of the surface area that is affected and the surface area that is unaffected. (i) Write down a differential equation for this model. (You are not required to solve your differential equation.) (2 marks) (ii) A solution of the differential equation for this model is given by A ¼ 9 1 þ 4e 0:09t Find the time taken for 50% of the area of the wall to be affected. Give your answer in hours to three significant figures. (4 marks) (16)

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18 (18)

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